Pub Date : 2024-09-09DOI: 10.1017/s001309152400035x
Guozheng Cheng, Xiang Fang, Chao Liu, Yufeng Lu
Motivated by new examples of functional Banach spaces over the unit disk, arising as the symbol spaces in the study of random analytic functions, for which the monomials ${z^n}_{ngeq 0}$ exhibit features of an unconditional basis yet they often don’t even form a Schauder basis, we introduce a notion called solid basis for Banach spaces and p-Banach spaces and study its properties. Besides justifying the rich existence of solid bases, we study their relationship with unconditional bases, the weak-star convergence of Taylor polynomials, the problem of a solid span and the curious roles played by c0. The two features of this work are as follows: (1) during the process, we are led to revisit the axioms satisfied by a typical Banach space of analytic functions over the unit disk, leading to a notion of $mathcal{X}^mathrm{max}$ (and $mathcal{X}^mathrm{min}$), as well as a number of related functorial constructions, which are of independent interests; (2) the main interests of solid basis lie in the case of non-separable (p-)Banach spaces, such as BMOA and the Bloch space instead of VMOA and the little Bloch space.
{"title":"Solid bases and functorial constructions for (p-)Banach spaces of analytic functions","authors":"Guozheng Cheng, Xiang Fang, Chao Liu, Yufeng Lu","doi":"10.1017/s001309152400035x","DOIUrl":"https://doi.org/10.1017/s001309152400035x","url":null,"abstract":"Motivated by new examples of functional Banach spaces over the unit disk, arising as the symbol spaces in the study of random analytic functions, for which the monomials <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S001309152400035X_inline1.png\"/> <jats:tex-math>${z^n}_{ngeq 0}$</jats:tex-math> </jats:alternatives> </jats:inline-formula> exhibit features of an unconditional basis yet they often don’t even form a Schauder basis, we introduce a notion called <jats:italic>solid basis</jats:italic> for Banach spaces and <jats:italic>p</jats:italic>-Banach spaces and study its properties. Besides justifying the rich existence of solid bases, we study their relationship with unconditional bases, the weak-star convergence of Taylor polynomials, the problem of a solid span and the curious roles played by <jats:italic>c</jats:italic><jats:sub>0</jats:sub>. The two features of this work are as follows: (1) during the process, we are led to revisit the axioms satisfied by a typical Banach space of analytic functions over the unit disk, leading to a notion of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S001309152400035X_inline2.png\"/> <jats:tex-math>$mathcal{X}^mathrm{max}$</jats:tex-math> </jats:alternatives> </jats:inline-formula> (and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S001309152400035X_inline3.png\"/> <jats:tex-math>$mathcal{X}^mathrm{min}$</jats:tex-math> </jats:alternatives> </jats:inline-formula>), as well as a number of related functorial constructions, which are of independent interests; (2) the main interests of solid basis lie in the case of non-separable (<jats:italic>p</jats:italic>-)Banach spaces, such as BMOA and the Bloch space instead of VMOA and the little Bloch space.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142193186","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-04DOI: 10.1017/s0013091524000245
Gonzalo Barranco Mendoza, Jawad Snoussi
We study pencils of curves on a germ of complex reduced surface $(S,0)$. These are families of curves parametrized by $ mathbb{P}^1 $ having 0 as the unique common point. We prove that for $win mathbb{P}^1$, the corresponding curve of the pencil does not have the generic topology if and only if either the corresponding curve of the pulled-back pencil to the normalized surface has a non generic topology or w is a limit value for the function $ f/g $ along the singular locus of $(S,0)$, where f and g are generators of the pencil.
我们研究复还原曲面$(S,0)$胚芽上的曲线铅笔。这些曲线是以mathbb{P}^1 $ 为参数、以 0 为唯一公共点的曲线族。我们证明，对于 $win mathbb{P}^1$，铅笔的相应曲线不具有泛函拓扑，当且仅当拉回铅笔到归一化曲面的相应曲线具有非泛函拓扑，或者 w 是函数 $ f/g $ 沿 $(S,0)$ 的奇点位置的极限值（其中 f 和 g 是铅笔的生成器）。
{"title":"Equisingularity in pencils of curves on germs of reduced complex surfaces","authors":"Gonzalo Barranco Mendoza, Jawad Snoussi","doi":"10.1017/s0013091524000245","DOIUrl":"https://doi.org/10.1017/s0013091524000245","url":null,"abstract":"<p>We study pencils of curves on a germ of complex reduced surface <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240603133454187-0844:S0013091524000245:S0013091524000245_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$(S,0)$</span></span></img></span></span>. These are families of curves parametrized by <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240603133454187-0844:S0013091524000245:S0013091524000245_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$ mathbb{P}^1 $</span></span></img></span></span> having 0 as the unique common point. We prove that for <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240603133454187-0844:S0013091524000245:S0013091524000245_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$win mathbb{P}^1$</span></span></img></span></span>, the corresponding curve of the pencil does not have the generic topology if and only if either the corresponding curve of the pulled-back pencil to the normalized surface has a non generic topology or <span>w</span> is a limit value for the function <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240603133454187-0844:S0013091524000245:S0013091524000245_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$ f/g $</span></span></img></span></span> along the singular locus of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240603133454187-0844:S0013091524000245:S0013091524000245_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$(S,0)$</span></span></img></span></span>, where <span>f</span> and <span>g</span> are generators of the pencil.</p>","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141258929","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-28DOI: 10.1017/s0013091524000324
Vincent Knibbeler, Sara Lombardo, Casper Oelen
We classify the automorphic Lie algebras of equivariant maps from a complex torus to $mathfrak{sl}_2(mathbb{C})$. For each case, we compute a basis in a normal form. The automorphic Lie algebras correspond precisely to two disjoint families of Lie algebras parametrised by the modular curve of $mathrm{PSL}_2({mathbb{Z}})$, apart from four cases, which are all isomorphic to Onsager’s algebra.
{"title":"A classification of automorphic Lie algebras on complex tori","authors":"Vincent Knibbeler, Sara Lombardo, Casper Oelen","doi":"10.1017/s0013091524000324","DOIUrl":"https://doi.org/10.1017/s0013091524000324","url":null,"abstract":"We classify the automorphic Lie algebras of equivariant maps from a complex torus to <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0013091524000324_inline1.png\"/> <jats:tex-math>$mathfrak{sl}_2(mathbb{C})$</jats:tex-math> </jats:alternatives> </jats:inline-formula>. For each case, we compute a basis in a normal form. The automorphic Lie algebras correspond precisely to two disjoint families of Lie algebras parametrised by the modular curve of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0013091524000324_inline2.png\"/> <jats:tex-math>$mathrm{PSL}_2({mathbb{Z}})$</jats:tex-math> </jats:alternatives> </jats:inline-formula>, apart from four cases, which are all isomorphic to Onsager’s algebra.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141165924","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-22DOI: 10.1017/s0013091524000208
Lucas Hall
Given a cocycle on a topological quiver by a locally compact group, the author constructs a skew product topological quiver and determines conditions under which a topological quiver can be identified as a skew product. We investigate the relationship between the ${C^*}$-algebra of the skew product and a certain native coaction on the ${C^*}$-algebra of the original quiver, finding that the crossed product by the coaction is isomorphic to the skew product. As an application, we show that the reduced crossed product by the dual action is Morita equivalent to the ${C^*}$-algebra of the original quiver.
{"title":"Coactions and skew products for topological quivers","authors":"Lucas Hall","doi":"10.1017/s0013091524000208","DOIUrl":"https://doi.org/10.1017/s0013091524000208","url":null,"abstract":"Given a cocycle on a topological quiver by a locally compact group, the author constructs a skew product topological quiver and determines conditions under which a topological quiver can be identified as a skew product. We investigate the relationship between the <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0013091524000208_inline1.png\"/> <jats:tex-math>${C^*}$</jats:tex-math> </jats:alternatives> </jats:inline-formula>-algebra of the skew product and a certain native coaction on the <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0013091524000208_inline2.png\"/> <jats:tex-math>${C^*}$</jats:tex-math> </jats:alternatives> </jats:inline-formula>-algebra of the original quiver, finding that the crossed product by the coaction is isomorphic to the skew product. As an application, we show that the reduced crossed product by the dual action is Morita equivalent to the <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0013091524000208_inline3.png\"/> <jats:tex-math>${C^*}$</jats:tex-math> </jats:alternatives> </jats:inline-formula>-algebra of the original quiver.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141153874","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-17DOI: 10.1017/s0013091524000373
Stefano Pinton, Peter Schlosser
This paper is inspired by a class of infinite order differential operators arising in quantum mechanics. They turned out to be an important tool in the investigation of evolution of superoscillations with respect to quantum fields equations. Infinite order differential operators act naturally on spaces of holomorphic functions or on hyperfunctions. Recently, infinite order differential operators have been considered and characterized on the spaces of entire monogenic functions, i.e. functions that are in the kernel of the Dirac operators. The focus of this paper is the characterization of infinite order differential operators that act continuously on a different class of hyperholomorphic functions, called slice hyperholomorphic functions with values in a Clifford algebra or also slice monogenic functions. This function theory has a very reach associated spectral theory and both the function theory and the operator theory in this setting are subjected to intensive investigations. Here we introduce the concept of proximate order and establish some fundamental properties of entire slice monogenic functions that are crucial for the characterization of infinite order differential operators acting on entire slice monogenic functions.
{"title":"Characterization of continuous homomorphisms on entire slice monogenic functions","authors":"Stefano Pinton, Peter Schlosser","doi":"10.1017/s0013091524000373","DOIUrl":"https://doi.org/10.1017/s0013091524000373","url":null,"abstract":"This paper is inspired by a class of infinite order differential operators arising in quantum mechanics. They turned out to be an important tool in the investigation of evolution of superoscillations with respect to quantum fields equations. Infinite order differential operators act naturally on spaces of holomorphic functions or on hyperfunctions. Recently, infinite order differential operators have been considered and characterized on the spaces of entire monogenic functions, i.e. functions that are in the kernel of the Dirac operators. The focus of this paper is the characterization of infinite order differential operators that act continuously on a different class of hyperholomorphic functions, called slice hyperholomorphic functions with values in a Clifford algebra or also slice monogenic functions. This function theory has a very reach associated spectral theory and both the function theory and the operator theory in this setting are subjected to intensive investigations. Here we introduce the concept of proximate order and establish some fundamental properties of entire slice monogenic functions that are crucial for the characterization of infinite order differential operators acting on entire slice monogenic functions.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141062253","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-16DOI: 10.1017/s0013091524000257
Dingxuan Tang, Zhiming Li
In the context of random amenable group actions, we introduce the notions of random upper metric mean dimension with potentials and the random upper measure-theoretical metric mean dimension. Besides, we establish a variational principle for the random upper metric mean dimensions. At the end, we study the equilibrium state for random upper metric mean dimensions.
{"title":"A variational principle of amenable random metric mean dimensions","authors":"Dingxuan Tang, Zhiming Li","doi":"10.1017/s0013091524000257","DOIUrl":"https://doi.org/10.1017/s0013091524000257","url":null,"abstract":"\u0000 In the context of random amenable group actions, we introduce the notions of random upper metric mean dimension with potentials and the random upper measure-theoretical metric mean dimension. Besides, we establish a variational principle for the random upper metric mean dimensions. At the end, we study the equilibrium state for random upper metric mean dimensions.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140970671","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-16DOI: 10.1017/s0013091524000348
Dalimil Peša
In this paper, we consider the question of smoothness of slowly varying functions satisfying the modern definition that, in the last two decades, gained prevalence in the applications concerning function spaces and interpolation. We show that every slowly varying function of this type is equivalent to a slowly varying function that has continuous classical derivatives of all orders.
{"title":"On the smoothness of slowly varying functions","authors":"Dalimil Peša","doi":"10.1017/s0013091524000348","DOIUrl":"https://doi.org/10.1017/s0013091524000348","url":null,"abstract":"In this paper, we consider the question of smoothness of slowly varying functions satisfying the modern definition that, in the last two decades, gained prevalence in the applications concerning function spaces and interpolation. We show that every slowly varying function of this type is equivalent to a slowly varying function that has continuous classical derivatives of all orders.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141062304","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-15DOI: 10.1017/s0013091524000336
LuÍs Carvalho, Cristina Diogo, Sérgio Mendes, Helena Soares
The numerical range in the quaternionic setting is, in general, a non-convex subset of the quaternions. The essential numerical range is a refinement of the numerical range that only keeps the elements that have, in a certain sense, infinite multiplicity. We prove that the essential numerical range of a bounded linear operator on a quaternionic Hilbert space is convex. A quaternionic analogue of Lancaster theorem, relating the closure of the numerical range and its essential numerical range, is also provided.
{"title":"On the convexity of the quaternionic essential numerical range","authors":"LuÍs Carvalho, Cristina Diogo, Sérgio Mendes, Helena Soares","doi":"10.1017/s0013091524000336","DOIUrl":"https://doi.org/10.1017/s0013091524000336","url":null,"abstract":"The numerical range in the quaternionic setting is, in general, a non-convex subset of the quaternions. The essential numerical range is a refinement of the numerical range that only keeps the elements that have, in a certain sense, infinite multiplicity. We prove that the essential numerical range of a bounded linear operator on a quaternionic Hilbert space is convex. A quaternionic analogue of Lancaster theorem, relating the closure of the numerical range and its essential numerical range, is also provided.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141064095","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-14DOI: 10.1017/s0013091524000361
Benjamin Martin
Let ${mathcal G}$ be a linear algebraic group over k, where k is an algebraically closed field, a pseudo-finite field or the valuation ring of a non-archimedean local field. Let $G= {mathcal G}(k)$. We prove that if $gammain G$ such that γ is a commutator and $deltain G$ such that $langle deltarangle= langle gammarangle$ then δ is a commutator. This generalises a result of Honda for finite groups. Our proof uses the Lefschetz principle from first-order model theory.
{"title":"Powers of commutators in linear algebraic groups","authors":"Benjamin Martin","doi":"10.1017/s0013091524000361","DOIUrl":"https://doi.org/10.1017/s0013091524000361","url":null,"abstract":"<p>Let <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240514121451157-0521:S0013091524000361:S0013091524000361_inline1.png\"><span data-mathjax-type=\"texmath\"><span>${mathcal G}$</span></span></img></span></span> be a linear algebraic group over <span>k</span>, where <span>k</span> is an algebraically closed field, a pseudo-finite field or the valuation ring of a non-archimedean local field. Let <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240514121451157-0521:S0013091524000361:S0013091524000361_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$G= {mathcal G}(k)$</span></span></img></span></span>. We prove that if <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240514121451157-0521:S0013091524000361:S0013091524000361_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$gammain G$</span></span></img></span></span> such that <span>γ</span> is a commutator and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240514121451157-0521:S0013091524000361:S0013091524000361_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$deltain G$</span></span></img></span></span> such that <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240514121451157-0521:S0013091524000361:S0013091524000361_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$langle deltarangle= langle gammarangle$</span></span></img></span></span> then <span>δ</span> is a commutator. This generalises a result of Honda for finite groups. Our proof uses the Lefschetz principle from first-order model theory.</p>","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140938305","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-08DOI: 10.1017/s0013091524000312
Augustin Mouze, Vincent Munnier
We are interested in the optimal growth in terms of Lp-averages of hypercyclic and $mathcal{U}$-frequently hypercyclic functions for some weighted Taylor shift operators acting on the space of analytic functions on the unit disc. We unify the results obtained by considering intermediate notions of upper frequent hypercyclicity between $mathcal{U}$-frequent hypercyclicity and hypercyclicity.
{"title":"Growth of hypercyclic functions: a continuous path between -frequent hypercyclicity and hypercyclicity","authors":"Augustin Mouze, Vincent Munnier","doi":"10.1017/s0013091524000312","DOIUrl":"https://doi.org/10.1017/s0013091524000312","url":null,"abstract":"We are interested in the optimal growth in terms of <jats:italic>L<jats:sup>p</jats:sup></jats:italic>-averages of hypercyclic and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0013091524000312_inline2.png\"/> <jats:tex-math>$mathcal{U}$</jats:tex-math> </jats:alternatives> </jats:inline-formula>-frequently hypercyclic functions for some weighted Taylor shift operators acting on the space of analytic functions on the unit disc. We unify the results obtained by considering intermediate notions of upper frequent hypercyclicity between <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" mimetype=\"image\" xlink:href=\"S0013091524000312_inline3.png\"/> <jats:tex-math>$mathcal{U}$</jats:tex-math> </jats:alternatives> </jats:inline-formula>-frequent hypercyclicity and hypercyclicity.","PeriodicalId":20586,"journal":{"name":"Proceedings of the Edinburgh Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140938304","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
$(S,0)$. These are families of curves parametrized by 
$ mathbb{P}^1 $ having 0 as the unique common point. We prove that for 
$win mathbb{P}^1$, the corresponding curve of the pencil does not have the generic topology if and only if either the corresponding curve of the pulled-back pencil to the normalized surface has a non generic topology or w is a limit value for the function 
$ f/g $ along the singular locus of 
$(S,0)$, where f and g are generators of the pencil.
${mathcal G}$ be a linear algebraic group over k, where k is an algebraically closed field, a pseudo-finite field or the valuation ring of a non-archimedean local field. Let 
$G= {mathcal G}(k)$. We prove that if 
$gammain G$ such that γ is a commutator and 
$deltain G$ such that 
$langle deltarangle= langle gammarangle$ then δ is a commutator. This generalises a result of Honda for finite groups. Our proof uses the Lefschetz principle from first-order model theory.
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